Tensor extrapolation methods with applications
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Tensor extrapolation methods with applications F. P. A. Beik1 · A. El Ichi2,3 · K. Jbilou3 · R. Sadaka4 Received: 13 April 2020 / Accepted: 10 September 2020 / © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract In this paper, we mainly develop the well-known vector and matrix polynomial extrapolation methods in tensor framework. To this end, some new products between tensors are defined and the concept of positive definitiveness is extended for tensors corresponding to T-product. Furthermore, we discuss on the solution of least-squares problem associated with a tensor equation using Tensor Singular Value Decomposition (TSVD). Motivated by the effectiveness of some proposed vector extrapolation methods in earlier papers, we describe how an extrapolation technique can be also implemented on the sequence of tensors produced by truncated TSVD (TTSVD) for solving possibly ill-posed tensor equations. Keywords Extrapolation · Ill-posed problems · Least-squares · Sequence of tensors · Tensor SVD · T-product Mathematics Subject Classification (2010) 65B05 · 15A69 · 65F22
1 Introduction In the last few years, several iterative methods have been proposed for solving large and sparse linear and nonlinear systems of equations. When an iterative process converges slowly, the extrapolation methods are required to obtain rapid convergence. The purpose of vector extrapolation methods is to transform a sequence of vector or matrices generated by some process to a new one that converges faster than the initial sequence. The well-known extrapolation methods can be classified into two categories: the polynomial methods that include the minimal polynomial extrapolation K. Jbilou
[email protected] 1
Department of Mathematics, Vali-e-Asr University of Rafsanjan, P.O. Box 518, Rafsanjan, Iran
2
Laboratory LABMIA-SI, Mohammed V University Rabat, Rabat, Morocco
3
Laboratory LMPA, University ULCO, 50 Rue F. Buisson, Calais cedex, France
4
Ecole Normale Superieure, Mohamed V University, Rabat, Morocco
Numerical Algorithms
(MPE) method of Cabay and Jackson [5], the modified minimal polynomial extrapolation (MMPE) method of Sidi, Ford, and Smith [27], the reduced rank extrapolation (RRE) method of Eddy [22] and Mesina [24], Brezinski [4] and Pugatchev [26]; and the -type algorithms including the topological -algorithm of Brezinski [4] and the vector -algorithm of Wynn [30]. Efficient implementations of some of these extrapolation methods have been proposed by Sidi [28] for the RRE and MPE methods using QR decomposition, while Jbilou and Sadok [12] give an efficient implementation of the MMPE based on a LU decomposition with pivoting strategy. It was also shown that when applied to linearly generated vector sequences, RRE and TEA methods are mathematically equivalent to GMRES and Lanczos methods, respectively. Those results were also extended to the block and global cases dealing with matrix sequences; see [11, 14]. Our aim in this paper is to define the analogue of these vector and matrix extrapo
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